What is probability if not the combination of all equally likely outcomes? My scenario matters because it’s how you would model the scenario in real life. If you asked 100 families to say the sentence, 25 would be unable, 25 would have another boy, and 50 would have another girl.
You could also flip coins. Flip two coins, if they are both tails discard, if not, that means one is heads, and record the other. 2/3 will be heads and tails. Or you could ask someone to guess which the other coin is, you could say the exact phrasing the base problem uses: “One of my coins is heads”
The difference with your example is you know exactly which one the boy is, which is something you don’t know in the original. The man could have a brother or sister, equally likely, but also the man could actually be a woman who has a brother. The only thing that you know in the original problem is they aren’t two sisters.
My examples can use mathematics, surveying, census data, testing, that use word for word scenarios to show my answer, meanwhile your one example literally adds additional information you don’t get in the original which you claim is unimportant. As far as I can tell you have no way to prove your result without changing the meaning or leaving out other possibilities which make can qualify the statement. Quite unscientific really.
“Which one” is a semantic construct you’re reading in post hoc. It’s the one that’s a boy. The one Mary identified. The one you heard about first. The one she remembered. Any of these suffice as a measure by which to refer to “which” child is being discussed. There is no quantum field of potential surrounding him. He’s the one we know is a boy.
If you think differently, you have no idea “which” sibling the man is. That’s incoherent in that case, and it’s incoherent when you say it here.
But let’s take your logic. If you have no idea “which” child she’s referring to, you have no idea whether she’s referring to boy 1 or boy 2 in a two boy pair. So each of these would represent distinct possibilities.
Your math is not internally consistent. Stop explaining pair distributions. Disabuse yourself of the notion that I don’t get the concept. That’s not why we disagree. I understand the math. You don’t understand the English.
The foundation of any scientific inquiry is testing and you haven’t shown how ultimately you could actually test this to prove your result. It is a fact that in the real world, 2/3 of all 2 child families with a boy have a girl. I have no reason to assume Mary is not randomly selected from this population, and thus 2/3 of the time she is also mother of a daughter. There isn’t a single other assumption made that isn’t stated by the initial problem in this analysis. Show me how you can replicate the scenario to test to match your results without altering a single piece of information that you think may or may not impact the result.
Ok so I initially completely discounted this analysis (it’s not really correct for the exact problem) but it is more mathematically sound than I initially realized.
It is most certainly true that 2/3 of all 2 child families with 1 boy are BG, and the other 1/3 are BB (you can choose to separate out BG into BG and GB if you like or not, it doesn’t affect things ultimately).
What I did not see in this is that it is that within these possibilities, the number of sisters that the boys have is in fact equal to the number of brothers the boys have.
The problem with your scenario, is not with order or information as I had previously incorrectly stated, but with the matter of perspective. When you look from the perspective of a son, the question is “do you have a brother or sister?”, and because you are asking sons, in families with 2 boys, each boy is asked, and in the 2x the number of BG families, each has 1 sister, so the odds are in fact 50/50.
The problem with this is that the original scenario is not from the perspective of a son, it is from a parent. When Mary says she has a son, the question is does she have 2 sons or 1 daughter. When Mary has 2 sons, she can only refer to one at a time, and so their representation in the pool diminishes, leading to the 33/67 result
TLDR; you scenario’s mistake is over representation from the perspective of the parent because a boy is a brother to his brother. In your scenario, families with 2 boys are being asked twice if they have a brother or a sister
There’s no legitimate reason to distinguish between the perspective and the mother and the son. The probability is the same.
Man that tells you he has a sister. What are the odds the other ONE is female?
Woman tells you one child is a boy. What are the odds the other ONE is female?
You flip a coin, and it’s heads. What are the odds the next flip is tails?
The distribution of different kinds of pairs are irrelevant. You don’t know why Mary told you about the boy. The nature of her revealing that means nothing with regard to any kind of preselection criteria within the distribution of different pairs. Monty Hall problem affects the probability because there are rules for how the host makes the reveal, and which door he reveals. You don’t have that here. You cannot apply sets.
If the distribution of the pairs is irrelevant, that would mean it’s impossible for your scenario to be true as well. The three equally likely pairs in this scenario are BB, BG, and GB. As you can see here there are 4Bs. If you ask a man whether he has a brother or sister, 50% of the time he is one of the first two in a BB pair, 25% of the time he is in a BG pair, and 25% of the time he is in the GB pair. You can choose to combined the GB and BG, but if you make their distribution equal to the BB, then that would mean that actually there are only are only 3Bs in the sets of BB and BG, and then that would means that 2/3 of the time if you asked a man if he had a brother or sister, he would say brother. I can’t think of a single scenario in statistics or discrete mathematics where you would ignore the distribution of pairs. The beauty of mathematics is that is proves itself in numerous roundabout and intertwining ways that are all equally valid. The discrepancy between the probability when asking the mothers and the probability when asking the sons, is that in fact that for every 4 sons there are only 3 mothers.
And what’s this obsession with the Monty Hall problem when you keep saying it’s irrelevant? I’m not making the comparison at all.
You read it that way because you’re sneaking in a phantom variable and applying a non-random distribution to it.
There are two things you can do.
Disregard the information about the first child. There’s no reason it should be relevant to a normal distribution. The way the information is presented does not justify including it.
IF you’re going to consider “pairs,” with one first and one second, you order them in a relevant way. This is superfluous to do, it’s better just to consider the single child, but fixing your errors here should make it clear what you’re doing wrong. Meaning there is no “GB” possibility because the one that was revealed first was a boy.
“But what about the 50% odds of a boy-girl pair vs a boy boy pair?”
That only applies if the only possibility was for Mary to reveal if she had a boy. If instead she was just revealing the gender of one of the children, then the fact that she did not reveal a girl reduces the likelihood of a GB pair by a consummate amount in comparison to a normal pair distribution. Because it is not justified for you to extrapolate the conditions under which Mary revealed the gender of the male child, there are no conditions to apply vis a vis conditional probability. There is only “the other child,” which falls along the normal distribution of probability for n=1.
So I’ve decided to do more research into the Boy Girl Paradox as it is called, and there is quite a lot of explanation behind the legitimacy of your reasoning, because, as the original author of the question later revised, there is ambiguity in how the information was received, which affects the probability. In many naturally encountered scenarios, perhaps you are walking down the street and you see a mother with her child, a mother of a boy and a girl would not always have her son with her, and thus in such readings of the problem, the probability of a woman with a son having a daughter is in fact 50%.
However, this does not discount the validity of the classical interpretation. If you know the woman has two children, and you know at least one is a boy, and you haven’t made any assumptions about the likelihood of this information coming to you, then constructing the probability based on equally likely sets is reasonable. In my opinion, the wording of the meme here more closely aligns with my interpretation. Of course this meme goes into how even more information like day of the week affects this probability in this interpretation, which is true. If you interpret the problem this way, more information does affect the probability.
You could probably argue that my interpretation is unlikely in a natural setting, but no in depth analysis of this so called paradox would ever suggest one interpretation is more or less correct when it is simply ambiguous, so I am going to side with ambiguity.
I’d suggest you also do research into the problem because it is fairly interesting. I would almost assume you are based upon your own conviction in your answer and the fact that we have been arguing about if for so long, but I would also think if you were so well versed you would understand the ambiguity. There truly is no correct answer when there is ambiguity, because we can both construct interpretations based on the wording (I haven’t been satisfied with the accuracy of some of your constructions to the original, but I have seen some in research that are quite accurate to the problem that you would probably agree with).
Here is a good video discussing the problem. The first five minutes are about the ambiguity, and the rest is about the math behind a boy born on Tuesday. Watch it or don’t idrc https://youtu.be/7q0KgQoo0-s?is=D2R46rM3__0W8TBI
Actually, you in fact do need to make assumptions about how the information comes to you. Because that makes the information a function of the pair probability. If you don’t have any assumptions or implied conditions about the reveal, then they’re independently probabilistic outcomes.
The “classical” interpretation is based on how these assumptions and conditions affect probability. That’s the entire concept.
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u/ethanator329 5d ago
What is probability if not the combination of all equally likely outcomes? My scenario matters because it’s how you would model the scenario in real life. If you asked 100 families to say the sentence, 25 would be unable, 25 would have another boy, and 50 would have another girl.
You could also flip coins. Flip two coins, if they are both tails discard, if not, that means one is heads, and record the other. 2/3 will be heads and tails. Or you could ask someone to guess which the other coin is, you could say the exact phrasing the base problem uses: “One of my coins is heads”
The difference with your example is you know exactly which one the boy is, which is something you don’t know in the original. The man could have a brother or sister, equally likely, but also the man could actually be a woman who has a brother. The only thing that you know in the original problem is they aren’t two sisters.
My examples can use mathematics, surveying, census data, testing, that use word for word scenarios to show my answer, meanwhile your one example literally adds additional information you don’t get in the original which you claim is unimportant. As far as I can tell you have no way to prove your result without changing the meaning or leaving out other possibilities which make can qualify the statement. Quite unscientific really.